Actually the topic. The new template did not add pagination output code to the delivery methods layout. Thus, if there are more than 400 delivery methods (the maximum possible number per page), then it is impossible to see those of them that go to the second and subsequent pagination pages (this is probably a rare problem, since few stores have more than 400 delivery options).
How to solve this:
At the end of the file:
/administrator/components/com_virtuemart/views/shipmentmethod/tmpl/default.php
Before the closing tag of table:
</table>
add the following code:
<tfoot>
<tr>
<td colspan="10">
<?php echo $this->pagination->getListFooter(); ?>
</td>
</tr>
</tfoot>
Hopefully this will be fixed in future updates.